Problem $2$:

A segway control system has transfer function

$P(s)=\frac{1}{s^2-49}$

The gyro sensor for the angle can be sampled as fast as $1$ ms and the loop computations require approximately $3$ ms $.$ Estimate approximately the constraints on the selection of the achievable crossover frequency with a strictly causal DT PID.

Use the same sampling rate, design a strictly causal DT PID controller for the maximum crossover frequency with PM $=45$deg.

Hint: Consider a PID where the time constant of the pseudo$-$differentiator is placed at Ts$/2$ so that, after Tustin discretization, it corresponds to an one$-$sample delay. Also recall that the ZOH can be approximated by a half$-$sample delay. Finally, the strictly causal PID implementation calls for an extra one$-$sample delay. On the other hand, the zeros of the PID are typically limited to a phase contribution of $75de\frac{g}{?}$ zero to maintain a reasonable integr$$l action $($i$.$e$.,$ avoid "canceling" the integrator$).$